Simplify the following expression and state the condition under which the simplification is valid: $y = \dfrac{n^2 + 10n + 9}{n^2 + 14n + 45}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{n^2 + 10n + 9}{n^2 + 14n + 45} = \dfrac{(n + 1)(n + 9)}{(n + 5)(n + 9)} $ Notice that the term $(n + 9)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(n + 9)$ gives: $y = \dfrac{n + 1}{n + 5}$ Since we divided by $(n + 9)$, $n \neq -9$. $y = \dfrac{n + 1}{n + 5}; \space n \neq -9$